\(P_{\mathcal {W}(V)} T|_{\mathcal {W}(V)}\)  is a pure contraction on \(\mathcal {W}(V)\) .
Statements \((i)\)  and \((ii)\)  are equivalent from [Proposition 3.1, [1]}]. From [Theorem 2.3, [2]}], we know that any \(n\) -tuple of doubly commuting pure isometries satisfies the wandering subspace property. The equivalence of \((i)\)  and \((iii)\)  now follows from this observation and Theorem REF , since an isometry is also a left-invertible operator by definition.

Wandering subspace for tuples of operators on tensor product of Hilbert spaces
For a \(n\) -tuple of left-invertible operators \(X=(X_1,\ldots ,X_n)\)  on \(\mathcal {H}\) , let us consider the tuple of left-invertible operators \(\mathcal {X}=(\mathcal {X}_1,\ldots , \mathcal {X}_n)\)  on \({H}:=\underset{n \text{ times}}{\underbrace{\mathcal {H}\otimes \ldots \otimes \mathcal {H}}}\)  defined by
\(\mathcal {X}_i = \underset{i-1 \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}} \otimes X_i \otimes \underset{n-i \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}}.\) 
It follows from a straightforward computation that the Cauchy dual of the left-invertible operator \(\mathcal {X}_i\)  is
\(\mathcal {X}_i^{\prime } = \underset{i-1 \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}} \otimes X_i^{\prime } \otimes \underset{n-i \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}}.\) 
The corresponding tuple of Cauchy duals is denoted by \(\mathcal {X}^{\prime } = (\mathcal {X}_1^{\prime } ,\ldots , \mathcal {X}_n^{\prime })\) . Now, let us compute the wandering subspace of these tuples of operators.
Lemma 3.7 
Let \(X=(X_1,\ldots ,X_n)\)  be a \(n\) -tuple of doubly commuting left-invertible operators on \(\mathcal {H}\) . Then \(\mathcal {W}(\mathcal {X}) = \mathcal {W}(\mathcal {X}^{\prime }) = \mathcal {W}(X_1) \otimes \ldots \otimes \mathcal {W}(X_n)\) .

Using the property of left-invertible operators we obtain that
\(P_{\ker \mathcal {X}_i^*} = \underset{i-1 \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}} \otimes P_{\ker X_i^*} \otimes \underset{n-i \text{ times}}{ \underbrace{I_{\mathcal {H}} \otimes \ldots \otimes I_{\mathcal {H}}}}.\) 
Note that \(\mathcal {W}(\mathcal {X}) = \bigcap _{i=1}^n \ker \mathcal {X}_i^* = \mathcal {W}(\mathcal {X}^{\prime })\) .
Since \(X\)  is a tuple of doubly commuting operators, both \(\mathcal {X}\)  and \(\mathcal {X}^{\prime }\) , are doubly commuting, which further implies that \((P_{\ker \mathcal {X}_1^*},\ldots ,P_{\ker \mathcal {X}_n^*})\)  is a commuting tuple of orthogonal projections and therefore,
\(P_{\mathcal {W}(\mathcal {X}^{\prime })} = P_{\bigcap _{i=1}^n \ker \mathcal {X}_i^*} = \prod _{i=1}^n P_{\ker \mathcal {X}_i^*} = P_{\ker X_1^*} \otimes \ldots \otimes P_{\ker X_n^*}.\) 
Thus,
\(\mathcal {W}(\mathcal {X}) = \bigcap _{i=1}^n \ker \mathcal {X}_i^* = \ker X_1^* \otimes \ldots \otimes \ker X_n^* = \mathcal {W}(X_1) \otimes \ldots \otimes \mathcal {W}(X_n).\) 
This completes the proof.
Theorem 3.8 
Let \(X=(X_1,\ldots ,X_n)\)  be a \(n\) -tuple of doubly commuting left-invertible operators on \(\mathcal {H}\)  such that both \(X_i\)  and \(X_i^{\prime }\)  possess the wandering subspace property on \(\mathcal {H}\)  for each \(i \in \lbrace 1,\ldots ,n\rbrace \) . Then both \(\mathcal {X}^{\prime }\)  and \(\mathcal {X}\)  possess the wandering subspace property on \({H}\) .

Our aim is to show that
\({H}= \bigvee _{\mathbf {m} \in \mathbb {N}^n} \mathcal {X}^{\prime \mathbf {m}} \mathcal {W}(\mathcal {X}^{\prime }).\) 
By Lemma REF , \(\mathcal {W}(\mathcal {X}^{\prime }) = \mathcal {W}(\mathcal {X}_1^{\prime }) \otimes \ldots \otimes \mathcal {W}(\mathcal {X}_n^{\prime })\) , which implies that
\(\mathcal {H}\otimes \mathcal {W}(X_2^{\prime }) \otimes \ldots \otimes \mathcal {W}(X_{n}^{\prime }) = \bigvee _{m \in \mathbb {N}^n} \mathcal {X}_1^{\prime m} \big (\mathcal {W}(\mathcal {X}_1^{\prime }) \otimes \ldots \otimes \mathcal {W}(\mathcal {X}_n^{\prime })) \big ) \subseteq \bigvee _{\mathbf {m} \in \mathbb {N}^n} \mathcal {X}^{\prime \mathbf {m}} \mathcal {W}(\mathcal {X}^{\prime }).\) 
Similarly,
\(\mathcal {H}\otimes \mathcal {H}\otimes \mathcal {W}(X_3^{\prime }) \otimes \ldots \otimes \mathcal {W}(X_{n}^{\prime }) = \bigvee _{m \in \mathbb {N}^n} \mathcal {X}_2^{\prime m} \big (\mathcal {H}\otimes \mathcal {W}(\mathcal {X}_2^{\prime }) \otimes \ldots \otimes \mathcal {W}(\mathcal {X}_n^{\prime }) \big ) \subseteq \bigvee _{\mathbf {m} \in \mathbb {N}^n} \mathcal {X}^{\prime \mathbf {m}} \mathcal {W}(\mathcal {X}^{\prime }).\) 
Following this process for another \((n-2)\)  times gives
\({H}\subseteq \bigvee _{\mathbf {m} \in \mathbb {N}^n} \mathcal {X}^{\prime \mathbf {m}} \mathcal {W}(\mathcal {X}^{\prime }) \subseteq {H}.\) 
This proves that \(\mathcal {X}^{\prime }\)  possesses the wandering subspace property on \({H}\) . Now \((\mathcal {X}^{\prime })^{\prime } = \mathcal {X}\)  and \(\mathcal {W}(X_i^{\prime }) = \mathcal {W}(X_i)\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \)  implies that \(\mathcal {X}\)  possesses the wandering subspace property on \({H}\) . This completes the proof.

Reproducing Kernel Hilbert spaces on the unit polydisc
Let \(\mathcal {H}_{\tilde{k}}\)  be the Hardy/
Bergman/Dirichlet space on unit disc \(\mathbb {D}\) , then \(M_z \in \mathcal {B}(\mathcal {H}_{\tilde{k}})\)  is a left-invertible operator. Moreover, [Proposition 2.7, [3]}] proves that both \(M_z\)  and the Cauchy dual \(M_z^{\prime }\)  possesses the wandering subspace property on \(\mathcal {H}_{\tilde{k}}\) . For a Hilbert space \(\mathcal {E}\) , let \(\mathcal {H}_{k} \otimes \mathcal {E}\)  denote the \(\mathcal {E}\) -valued Hardy/Bergman/Dirichlet kernel on \(n\)  that is, \(k(\mathbf {z}, \mathbf {w}) = \Pi _{i=1}^n {\tilde{k}}(z_i, \mathbf {w_i})\)  (\(\mathbf {z}, \mathbf {w} \in n\) ). It is well known that the map
\(U: \mathcal {H}_{k}(n) \times \mathcal {E}\rightarrow (\underset{n\text{-times}}{\underbrace{\mathcal {H}_{\tilde{k}}( \otimes \ldots \otimes \mathcal {H}_{\tilde{k}}(}}) \otimes \mathcal {E},\) 
defined by \( U(\mathbf {z}^{\mathbf {m}} \otimes \eta ) = (z_1^{m_1} \otimes \ldots \otimes z_n^{m_n}) \otimes \eta \quad (\mathbf {z} \in n, \mathbf {m} \in \mathbb {N}^n, \eta \in \mathcal {E}),\)  is a unitary and moreover, \(U (M_{z_i} \otimes I_{\mathcal {E}}) = (\tilde{M}_{z_i} \otimes I_{\mathcal {E}}) U\) , where \(\tilde{M}_{z_i} = \underset{i\text{-times}}{\underbrace{I_{\mathcal {H}_{\tilde{k}}} \otimes \ldots \otimes I_{\mathcal {H}_{\tilde{k}}}}} \otimes M_z \otimes \underset{(n-i)\text{-times}}{\underbrace{I_{\mathcal {H}_{\tilde{k}}} \otimes \ldots \otimes I_{\mathcal {H}_{\tilde{k}}}}}\) . Using this identification and Theorem REF , we can deduce the following result.
Theorem 3.9 
For a Hilbert space \(\mathcal {E}\) , let \(\mathcal {H}_k(n) \otimes \mathcal {E}\)  be the \(\mathcal {E}\) -valued Hardy/Bergman/ Dirichlet space on the polydisc. Then both \(\tilde{M}_{\mathbf {z}}^{\prime }:=(\tilde{M}_{z_1}^{\prime } \otimes I_{\mathcal {E}}, \ldots ,\tilde{M}_{z_n}^{\prime } \otimes I_{\mathcal {E}})\)  and \(\tilde{M}_{\mathbf {z}}:=(\tilde{M}_{z_1} \otimes I_{\mathcal {E}}, \ldots ,\tilde{M}_{z_n} \otimes I_{\mathcal {E}})\)  possess the wandering subspace property on \(\mathcal {H}_k(n) \otimes \mathcal {E}\) .

We are now ready to prove the main result of this section.
Note that \(\Phi (\mathbf {z}) \in \mbox{Mult} (\mathcal {H}_k(n) \otimes \mathcal {E})\)  implies that \(\Phi (\mathbf {z}) \in H_{\mathcal {B}(\mathcal {E})}^{\infty }(n)\)  and therefore, \(M_{\Phi } (M_{z_i} \otimes I_{\mathcal {E}}) = (M_{z_i} \otimes I_{\mathcal {E}}) M_{\Phi }\)  for \(i \in \lbrace 1,\ldots ,n\rbrace \) . This implies that \((\tilde{M}_{z_i} \otimes I_{\mathcal {E}}) U M_{\Phi } U^* = U M_{\Phi } U^* (\tilde{M}_{z_i} \otimes I_{\mathcal {E}})\) . Now, using Theorem REF  and Theorem REF  we obtain that \(U M_{\Phi } U^*\)  is a pure contraction on \((\mathcal {H}_{\tilde{k}}( \otimes \ldots \otimes \mathcal {H}_{\tilde{k}}() \otimes \mathcal {E}\)  if and only if \(P_{\mathcal {W}(\tilde{M}_{\mathbf {z}})} (U M_{\Phi } U^*)|_{\mathcal {W}(\tilde{M}_{\mathbf {z}})}\)  is a pure contraction on \(\mathcal {W}(\tilde{M}_{\mathbf {z}})\) . By definition \(U^* \mathcal {W}(\tilde{M}_{\mathbf {z}}) = \mathcal {W}(M_{\mathbf {z}}) = \mathcal {E}\) , which implies that \(P_{\mathcal {W}(\tilde{M}_{\mathbf {z}})} (U M_{\Phi } U^*)|_{\mathcal {W}(\tilde{M}_{\mathbf {z}})}\)  is a pure contraction on \(\mathcal {W}(\tilde{M}_{\mathbf {z}})\)  if and only if \(\Phi (0)\)  is a pure contraction on \(\mathcal {E}\) . This completes the proof.
In the scalar-valued spaces, we have the following result.
Theorem 3.10 
Let \(\mathcal {H}_k\)  be the scalar-valued Hardy/Bergman/Dirichlet space on \(n\) . Then for any \(\phi (\mathbf {z}) \in \mbox{Mult}_{1} (\mathcal {H}_k)\) , the following statements are equivalent:

\(M_{\Phi }\)  is a pure contraction on \(\mathcal {H}_K\) .

\(|\Phi (0)|<1\) .


Thus, we obtain [Theorem 3.2, [4]}] as a corollary of our results. In the case of inner multipliers of the Hardy space on polydisc, our methods give an alternative proof of following result observed in [Theorem 4.1, [5]}].
Corollary 3.11 Let \(\theta (\mathbf {z}) \in H^{\infty }(n)\)  be a non-constant inner function (that is, \(|\theta (\mathbf {z})| = 1\)  a.e. on \(\mathbb {T}^n\) ). Then \(M_{\theta }\)  is always a pure isometry on \(H^2(n)\) .

By the maximum principle, \(|\theta (0)|<1\)  for any non-constant inner function \(\theta (\mathbf {z}) \in H^{\infty }(n)\) . Now, the proof follows by applying Theorem REF  for \(H^2(n)\) .
Remark 3.3 From our discussion on left-invertibility of shift operators in the Section , it follows that Theorem REF  can be extended for weighted Bergman spaces \(A_{\alpha , \mathcal {E}}(\mathbb {D}^n)\) , where \(-1 < \alpha \le 0\) , as well.

We are now interested in finding a result involving the restriction of mulitplication operators on certain \(M_{\mathbf {z}}\) -joint invariant subspaces of the \(H^2(n)\) . For proving this result, let us first recall that a \(M_{\mathbf {z}}\) -joint invariant closed subspace of \(H^2(n)\)  is said to be doubly commuting if the commutator \([R_{z_i}^*,R_{z_j}] = 0\)  for \(i,j \in \lbrace 1,\ldots ,n\rbrace \)  such that \(i \ne j\) , where \(R_{z_i} := M_{z_i}|_{\mathcal {S}}\) . By [Theorem 1.3, [6]}] any such \(\mathcal {S}\)  is of the form \(\theta H^2(n)\)  for some inner function \(\theta \in H^{\infty }(n)\) . For the purpose of the next result, we shall call \(\mathcal {S}= \theta H^2(\mathbb {D}^n)\)  to be a non-zero based doubly commuting \(M_{\mathbf {z}}\) -joint invariant subspace of \(H^2(\mathbb {D}^n)\) , if \(\theta (0) \ne 0\) .
Theorem 3.12 
Let \(\phi \in H^{\infty }(n)\)  be a contractive multiplier. Then \(M_{\phi }\)  is a pure contraction on \(H^2(n)\)  if and only if there exists a non-zero based doubly commuting \(M_{\mathbf {z}}\) -joint invariant subspace of \(H^2(\mathbb {D}^n)\)  such that \(M_{\phi }|_{\mathcal {S}}\)  is a pure contraction.

Observe that \(M_{\phi }\)  is a pure contraction on \(H^2(n)\)  will always imply \(M_{\phi }|_{\mathcal {S}}\)  is a pure contraction for any \(M_{\mathbf {z}}\) -joint invariant closed subspace \(\mathcal {S}\) . Now, if there exists such a subspace \(\mathcal {S}\)  for which \(M_{\phi }|_{\mathcal {S}}\)  is a pure contraction, then for any \(f \in \mathcal {S}\)  we have
\(\lim _{n \rightarrow \infty } \Vert P_{\mathcal {S}} M_{\Phi }^{*n} f\Vert  = 0.\) 
In particular, for \(f(z) = P_{\mathcal {S}} k_{0}(\mathbf {z}) = P_{\mathcal {S}}1\)  (note that \(\theta (0) \ne 0\)  shows that \(P_{\mathcal {S}}1 \ne 0\) ) we have
\(\Vert P_{\mathcal {S}} M_{\phi }^{*n} P_{\mathcal {S}}1\Vert  = \Vert M_{\theta } M_{\theta }^* M_{\phi }^{*n} M_{\theta } M_{\theta }^* k_{0}(\mathbf {z}) \Vert  = \Vert \overline{\theta (0)} M_{\phi }^{*n} k_{0}(\mathbf {z})\Vert  = |\overline{\theta (0)} \hspace{2.84526pt}\overline{\phi (0)}^{n}|.\) 
This implies that \(|\overline{\phi (0)}^{n}| \rightarrow 0\)  as \(n \rightarrow \infty \)  and therefore, by Corollary REF , it follows that \(M_{\Phi }\)  is a pure contraction on \(H^2(n)\) . This completes the proof.

Pure contractive multipliers on the unit ball
In this section, we obtain a characterization for pure contractive multipliers of several reproducing kernel Hilbert spaces on \(\mathbb {B}_n\) . For this purpose, we are interested in the following type of rkHs's.
Definition 4.1 A \(\mathcal {E}\) -valued reproducing kernel Hilbert space \(\mathcal {H}_k \otimes \mathcal {E}\)  on \(\mathbb {B}_n\)  is said to be regular if it satisfies the following conditions:
\((i) \hspace{2.84526pt}\mathcal {H}_k \otimes \mathcal {E}\subset \mathcal {O}(\mathbb {B}_n, \mathcal {E})\) ;
\((ii) \hspace{2.84526pt} M_{\mathbf {z}}= (M_{z_1},\ldots ,M_{z_n})\)  is a tuple of bounded operators on \(\mathcal {H}_k \otimes \mathcal {E}\) ;
\((iii)\)  the collection of monomials form an orthogonal basis of \(\mathcal {H}_k \otimes \mathcal {E}.\)
\((iv)\)  \(\bigcap _{i=1}^n \ker M_{z_i}^* = \mathcal {E}\) .

The author is grateful to Prof. Kunyu Guo at the Fudan University, China for a correspondence that led to establishing the following result.
Proposition 4.2 
Let \(\mathcal {H}_k \otimes \mathcal {E}\)  be a \(\mathcal {E}\) -valued regular rkHs on \(\mathbb {B}_n\)  and let \(\mathcal {M}\)  be a \(M_{\mathbf {z}}\) -joint invariant closed subspace of \(\mathcal {H}_k \otimes \mathcal {E}\) . Then \(\mathcal {M}\ne \lbrace 0\rbrace \)  if and only if \(\mathcal {W}(M_{\mathbf {z}}|_{\mathcal {M}}) \ne \lbrace 0\rbrace \) .

It is easy to observe that \(\bigcap _{l=0}^{\infty } \sum _{|\mathbf {m}| = l} M_{\mathbf {z}}^{\mathbf {m}} \mathcal {H}_k \otimes \mathcal {E}= \lbrace 0\rbrace \) . Otherwise, the order at zero for any function \(f \in \bigcap _{l=0}^{\infty } \sum _{|\mathbf {m}| = l} M_{\mathbf {z}}^{\mathbf {m}} \mathcal {H}_k \otimes \mathcal {E}\) , can be increased arbitrarily and this is possible only when \(f = 0\) . Thus, \(M_{\mathbf {z}}\)  satisfies the assumption of Proposition REF  and therefore, we get that \(\mathcal {W}(M_{z}|_{\mathcal {M}}) \ne \lbrace 0\rbrace \)  if and only if \(\mathcal {M}\ne \lbrace 0\rbrace \) . This completes the proof.
In the setting of the unit ball, we will now prove a result that is analogous to Lemma REF .
Lemma 4.3 
Let \(\mathcal {H}_k \otimes \mathcal {E}\)  be a \(\mathcal {E}\) -valued regular rkHs on \(\mathbb {B}_n\)  such that

\(k/s \ge 0\) , where \(s\)  is a cnp kernel normalized at 0;

\(\mathcal {H}_s\)  is a regular rkHs on \(\mathbb {B}_n\) .

Let \(\mathcal {M}\)  be a proper \(M_{\mathbf {z}}\) -invariant closed subspace of \(\mathcal {H}_k \otimes \mathcal {E}\)  such that \(\mathcal {W}(M_{\mathbf {z}}) \subseteq \mathcal {M}^{\perp }\) . Then there exists a non-zero vector \(\eta \in \mathcal {M}\)  such that \(M_{z_i}^* \eta \in \mathcal {M}^{\perp }\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .

At first, let us observe that \(\mathcal {M}\)  cannot be a \(M_{\mathbf {z}}\) -joint reducing subspace of \(\mathcal {H}_k \otimes \mathcal {E}\)  as otherwise, the assumption \(\mathcal {W}(M_{\mathbf {z}}) = \mathcal {E}\subseteq \mathcal {M}^{\perp }\)  will contradict that \(\mathcal {M}\)  is a proper subspace.
From Theorem REF , it follows that there exists a Hilbert space \(\mathcal {F}\)  and a partially isometric multiplier \(\Gamma (\mathbf {z}) \in \mbox{Mult}(\mathcal {H}_s \otimes \mathcal {F}, \mathcal {H}_k \otimes \mathcal {E})\)  such that
\(\mathcal {M}^{\perp } = \mathcal {H}_k \otimes \mathcal {E}\ominus \Gamma (\mathbf {z}) \big (\mathcal {H}_s \otimes \mathcal {F}\big ).\) 
The proof can be divided into the following cases.
Case (i): If there exists a non-zero vector \(\eta \in \operatorname{ran}M_{\Gamma }^* \cap \mathcal {F}\) , then for any \(f \in \mathcal {H}_s \otimes \mathcal {F}\)  and \(i \in \lbrace 1,\ldots ,n\rbrace \) , we have
\(\langle M_{z_i}^* M_{\Gamma } \eta , M_{\Gamma } f \rangle = \langle M_{\Gamma } \eta , M_{z_i} M_{\Gamma } f \rangle = \langle M_{\Gamma }^* M_{\Gamma } \eta , M_{z_i} f \rangle = \langle \eta , M_{z_i} f \rangle =0.\) 

This implies that \(M_{z_i}^* M_{\Gamma }\eta \in \mathcal {M}^{\perp }\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .
Case (ii): We will obtain a contradiction in the case of \(\operatorname{ran}M_{\Gamma }^*\cap \mathcal {F}= \lbrace 0\rbrace \) . From equation (REF ) we get that \(M_{\Gamma }: \mathcal {H}_s \otimes \mathcal {F}\rightarrow \mathcal {M}\)  is a partial isometry and moreover, \(\Gamma (\mathbf {z})\)  is a multiplier implies that \(M_{\Gamma } M_{z_i} = M_{z_i}|_{\mathcal {M}} M_{\Gamma }\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) . Now applying Lemma REF  gives
\(\mathcal {W}(M_{\mathbf {z}}|_{\mathcal {M}}) = M_{\Gamma } (\operatorname{ran}M_{\Gamma }^*\cap \mathcal {F}) = \lbrace 0\rbrace .\) 
From Proposition REF  we have \(\mathcal {M}= \lbrace 0\rbrace \) , which is a contradiction to our assumption that \(\mathcal {M}\)  is proper. This completes the proof.
We are now ready to establish our main result for rkHs's \(H_m(\mathbb {B}_n,\mathcal {E})\) , for integers \(m \ge 1\) . First, let us note the following operator identity that has been established in [Lemma 2, [7]}].
\(P_{\mathcal {E}} = I_{H_m(\mathbb {B}_n, \mathcal {E})} - \sum _{j=0}^{m-1} (-1)^j {m \atopwithdelims ()j+1} \sum _{| \alpha | = j+1} \gamma _{\alpha } M_{\mathbf {z}}^{\alpha } M_{\mathbf {z}}^{*\alpha } \quad (\alpha \in \mathbb {N}^n),\) 
where \(\gamma _{\alpha }\)  is the coefficient appearing in the following sum
\(K_m^{-1}(\mathbf {z},\bar{\mathbf {w}}) = (1 - \langle \mathbf {z}, \bar{\mathbf {w}} \rangle )^{m} I_{\mathcal {E}} = \sum _{j=0}^m (-1)^j {m \atopwithdelims ()j} \sum _{|\alpha | = j}\gamma _{\alpha } \mathbf {z}^{\alpha } \mathbf {w}^{\alpha } I_{\mathcal {E}} \quad (\mathbf {z},\mathbf {w} \in \mathbb {B}_n).\) 
Note that \(m \ge 1\)  implies that the kernel \(K_m\)  of the rkHs \(H_m(\mathbb {B}_n,\mathcal {E})\)  always admits a cnp factor \(s = (1 - \langle \mathbf {z}, \mathbf {w} \rangle )^{-1}\) , that is the kernel for the Drury-Arveson space.
Theorem 4.4 
Let \(\Phi \in \mbox{Mult}_1(H_m(\mathbb {B}_n,\mathcal {E}))\) , where \(m \ge 1\)  (and \(m \in \mathbb {N}\) ). Then the following are equivalent

\(M_{\Phi }\)  is a pure contraction on \(H_m(\mathbb {B}_n,\mathcal {E})\) ,

\(\Phi (0)\)  is a pure contraction on \(\mathcal {E}\) .


\((i)\)  implies \((ii)\)  is easy to observe since for any \(\eta \in \mathcal {E}\)  we have
\(\lim _{l \rightarrow \infty } \Vert \Phi (0)^{*l} \eta \Vert  = \lim _{l \rightarrow \infty } \Vert M_{\Phi }^{*l} K_m(\cdot ,0) \eta \Vert  = 0.\) 
\((ii) \Rightarrow (i)\) : It is easy to observe that \(\ker A_{M_{\Phi }}\)  is a \((M_{z_1}^*,\ldots ,M_{z_n}^*)\) -joint invariant closed subspace of \(H_m(\mathbb {B}_n, \mathcal {E})\) . As in the proof of Theorem REF , let us focus on the case where \(\ker A_{M_{\Phi }}\)  is not a \(M_{\mathbf {z}}\) -joint reducing subspace of \(H_m(\mathbb {B}_n,\mathcal {E})\) . Furthermore, if we assume that \(\ker A_{M_{\Phi }}\)  is a proper subspace, then from Lemma REF , there exists a non-zero element \(\eta \in (\ker A_{M_{\Phi }})^{\perp }\)  such that \(M_{z_i}^* \eta \in \ker A_{M_{\Phi }}\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) . Now, for any \(l \in \mathbb {N}\) , there exists \(\zeta _l \in \mathcal {E}\)  and \(h_{j,l} \in z_j \mathcal {H}\)  such that
\(M_{\Phi }^{*l} \eta = \zeta _l \oplus \sum _{t=1}^n h_{t,l},\) 
and therefore, \(\lim _{l \rightarrow \infty } \Vert M_{z_i}^* \sum _{t=1}^n h_{t,l} \Vert  = \lim _{l \rightarrow \infty } \Vert M_{z_i}^* M_{\Phi }^{*l} \eta \Vert  = \lim _{l \rightarrow \infty } \Vert M_{\Phi }^{*l} M_{z_i}^* \eta \Vert  = 0\) , for any \(i \in \lbrace 1,\ldots ,n\rbrace \) . Thus,
\(\lim _{l \rightarrow \infty } \Vert \sum _{j=0}^{m-1} (-1)^j {m \atopwithdelims ()j+1} \sum _{|\alpha | = j+1} \gamma _{\alpha } M_{\mathbf {z}}^{\alpha } M_{\mathbf {z}}^{*\alpha } (\sum _{t=1}^n h_{t,l} )\Vert  =0.\) 
From equation (REF ), we know that
\(\sum _{j=0}^{m-1} (-1)^j {m \atopwithdelims ()j+1} \sum _{|\alpha | = j+1} \gamma _{\alpha } M_{\mathbf {z}}^{\alpha } M_{\mathbf {z}}^{*\alpha } = I_{H_m(\mathbb {B}_n, \mathcal {E})} - P_{\mathcal {E}},\) 
and therefore,
\(\lim _{l \rightarrow \infty } \Vert  \sum _{t=1}^n h_{t,l} \Vert  &= \lim _{l \rightarrow \infty } \Vert (I_{H_m(\mathbb {B}, \mathcal {E})} - P_{\mathcal {E}}) (\sum _{t=1}^n h_{t,l} ) \Vert  \\&= \lim _{l \rightarrow \infty } \Vert \sum _{j=0}^{m-1} (-1)^j {m \atopwithdelims ()j+1} \sum _{|\alpha | = j+1} \gamma _{\alpha } M_{\mathbf {z}}^{\alpha } M_{\mathbf {z}}^{*\alpha } (\sum _{t=1}^n h_{t,l} )\Vert  = 0.\) 

Now,
\(\Vert A_{M_{\Phi }} \eta \Vert  = \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }} M_{\Phi }^{*l} M_{\Gamma } (1 \otimes \eta )\Vert = \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }}(\zeta _l \oplus \sum _{t=1}^n h_{t,l})\Vert &= \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }}(\sum _{t=1}^n h_{t,l})\Vert  \\& \le \lim _{l \rightarrow \infty } \Vert \sum _{t=1}^n h_{t,l}\Vert  = 0,\) 

implies that \(\eta \in \ker A_{M_{\Phi }}\) . But \(\eta \)  was assumed to be an element in \((\ker A_{M_{\Phi }})^{\perp }\)  which is possible only if \(\eta = 0\) . This is a contradiction to our assumption that \(\ker A_{M_{\Phi }}\)  is a proper subspace. Since \(\lbrace 0\rbrace  \ne \mathcal {E}\subset \ker A_{M_{\Phi }}\) , the only possibility is that \(\ker A_{M_{\Phi }} = H_m(\mathbb {B}_n, \mathcal {E})\) . In other words, \(M_{\Phi }\)  is a pure contraction on \(H_m(\mathbb {B}_n, \mathcal {E})\) . This completes the proof.
We can establish the above result for unitarily-invariant cnp spaces as well. Similar to the above situation, we require the following identity analogous to equation (REF ). This result was established by Chen in [Proposition 2.1, Lemma 2.2, [8]}].
Proposition 4.5 (Chen) 
Let \(\mathcal {H}_k \otimes \mathcal {E}\)  be an unitarily invariant complete Nevanlinna-Pick space on \(\mathbb {B}_n\) . Then there exist real numbers \(c_{|\beta |}\)  such that \(c_{|\beta |} \le 0\)  for \(|\beta | \ge 1\)  and for which
\(k^{-1}(M_{\mathbf {z}}, M_{\mathbf {z}}^*) = \sum _{|\beta |=0}^{\infty } c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } = P_{\mathcal {E}} \quad (\beta \in \mathbb {N}^n).\) 

Note that for any \(h \in \mathcal {H}_k \otimes \mathcal {E}\)  and \(n \ge 2\)  we have
\(\langle \sum _{1 \le |\beta | \le n} c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } h, h \rangle - \langle \sum _{1 \le |\beta | \le n+1} c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } h, h \rangle &= - \langle \sum _{|\beta | = n+1} c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } h, h \rangle \\&= \sum _{|\beta | = n+1} - c_{|\beta |} \frac{|\beta | !}{\beta !} \Vert  M_{\mathbf {z}}^{*\beta } h \Vert ^2 \\& \ge 0.\) 

Thus, \(\lbrace  \langle \underset{1 \le |\beta | \le n}{\sum }c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } h, h \rangle \rbrace _n\)  is a monotonic decreasing sequence of real numbers.
Theorem 4.6 
Let \(\mathcal {H}_k \otimes \mathcal {E}\)  be an unitarily invariant complete Nevanlinna-Pick space on \(\mathbb {B}_n\) . Then for any \(\Phi \in \mbox{Mult}_1(\mathcal {H}_k \otimes \mathcal {E})\) , the following are equivalent

\(M_{\Phi }\)  is a pure contraction on \(\mathcal {H}_k \otimes \mathcal {E}\) ,

\(\Phi (0)\)  is a pure contraction on \(\mathcal {E}\) .


As before, \((i)\)  implies \((ii)\)  is easy to observe. We will prove \((ii)\)  implies \((i)\) .
Note that \(\ker A_{M_{\Phi }}\)  is a \((M_{z_1}^*,\ldots ,M_{z_n}^*)\) -joint invariant closed subspace of \(\mathcal {H}_k \otimes \mathcal {E}\) . Similar to the proof of Theorem REF , let us consider the case where \(\ker A_{M_{\Phi }}\)  is a proper subspace that is not \(M_{\mathbf {z}}\) -joint reducing. Then by Lemma REF  there exists \(\eta \in (\ker A_{M_{\Phi }})^{\perp }\)  such that \(M_{z_i}^* \eta \in \ker A_{M_{\Phi }}\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) . Now for any \(l \in \mathbb {N}\) , there exists \(\zeta _l \in \mathcal {E}\)  and \(h_{j,l} \in z_j \mathcal {H}\)  such that
\(M_{\Phi }^{*l} \eta = \zeta _l \oplus \sum _{j=1}^n h_{j,l},\) 
Thus, \(\lim _{l \rightarrow \infty } \Vert M_{z_i}^* \sum _{j=1}^n h_{j,l} \Vert  = 0\)  for any \(i \in \lbrace 1,\ldots ,n\rbrace \)  which further implies that
\(\lim _{l \rightarrow \infty } \Vert \sum _{|\beta | = 1} c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta }(\sum _{j=1}^n h_{j,l} )\Vert  =0.\) 
From Proposition REF  it follows that
\(\sum _{|\beta |=1}^{\infty } c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta } = I_{\mathcal {H}_k \otimes \mathcal {E}} - P_{\mathcal {E}},\) 
and thus,
\(\lim _{l \rightarrow \infty } \Vert  \sum _{j=1}^n h_{j,l} \Vert  = \lim _{l \rightarrow \infty } \langle \sum _{j=1}^n h_{j,l}, \sum _{j=1}^n h_{j,l} \rangle &= \lim _{l \rightarrow \infty } \langle (I_{\mathcal {H}_k \otimes \mathcal {E}} - P_{\mathcal {E}}) (\sum _{j=1}^n h_{j,l} ), \sum _{j=1}^n h_{j,l} \rangle \\&=\lim _{l \rightarrow \infty } \langle \sum _{|\beta |=1}^{\infty } c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta }(\sum _{j=1}^n h_{j,l} ), \sum _{j=1}^n h_{j,l} \rangle \\&= \lim _{l \rightarrow \infty } \lim _{m \rightarrow \infty } \langle \sum _{1 \le |\beta | \le m}c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta }(\sum _{j=1}^n h_{j,l} ), \sum _{j=1}^n h_{j,l}\rangle \\& \le \lim _{l \rightarrow \infty } \langle \sum _{|\beta | = 1} c_{|\beta |} \frac{|\beta | !}{\beta !} M_{\mathbf {z}}^{\beta } M_{\mathbf {z}}^{*\beta }(\sum _{j=1}^n h_{j,l} ), \sum _{j=1}^n h_{j,l}\rangle \\&= 0 \quad (\text{since } \Vert \sum _{j=1}^n h_{j,l}\Vert  \le \eta \text{, for all } l \in \mathbb {N}).\) 

Now,
\(\Vert A_{M_{\Phi }} \eta \Vert  = \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }} M_{\Phi }^{*l} \eta \Vert = \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }}(\zeta _l \oplus \sum _{j=1}^n h_{j,l})\Vert &= \lim _{l \rightarrow \infty } \Vert A_{M_{\Phi }}(\sum _{j=1}^n h_{j,l})\Vert  \\& \le \lim _{l \rightarrow \infty } \Vert \sum _{j=1}^n h_{j,l}\Vert  = 0,\) 

This implies that \(\eta \in \ker A_{M_{\Phi }}\)  and therefore, the only possibility is \(\eta = 0\) . This is a contradiction to our assumption that \(\ker A_{M_{\Phi }}\)  is proper which further implies that \(\ker A_{M_{\Phi }} = \mathcal {H}_k \otimes \mathcal {E}\) , in other words, \(M_{\Phi }\)  is a pure contraction on \(\mathcal {H}_k \otimes \mathcal {E}\) .
Using Theorem REF , we can establish a result that is analogous to Theorem REF , with respect to certain special tuples of commuting operators. These tuples are indeed special because in a recent paper [9]}, Eschmeier and Langendörfer established a Wold-von Neumann type decomposition theorem for these tuples. Let us give a brief overview of these tuples before moving into our results (see [9]} for more details).
Let \(X\)  be the row operator (denoted by \(X \in \mathcal {B}(\mathcal {H})^n\) ) corresponding to a \(n\) -tuple of commuting operators \((X_1,\ldots ,X_n)\)  on \(\mathcal {H}\) . Let \(\sigma _X: \mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathcal {H})\)  be a positive linear map defined by
\(\sigma _X(Y): = \sum _{1 \le i \le n} X_i Y X_i^* \quad (Y \in \mathcal {B}(\mathcal {H})).\) 
\(X\)  is said to be pure if \(\sigma _X^n(I) \rightarrow 0\)  in the strong operator topology as \(n \rightarrow \infty \) .
Definition 4.7 A row operator \(X \in \mathcal {B}(\mathcal {H})^n\)  is said to be regular at \(z=0\)  if there is a real number \(\epsilon > 0\)  such that \(\Vert z\Vert  < \epsilon \) , the subspace \((X - Z) \mathcal {H}^n \subset \mathcal {H}\)  is closed and \(\mathcal {H}\)  decomposes into the algebraic direct sum
\(\mathcal {H}= (X - Z) \mathcal {H}^n \oplus \mathcal {W}(X),\) 
where \(Z \in \mathcal {B}(\mathcal {H})^n\)  is the row operator defined by \(Z (h_i)_{i=1}^n = \sum _{i=1}^n z_i h_i\) .

The following result is a particular case of [Theorem 3.7, Corollary 3.6, [9]}].
Theorem 4.8 Let \(X \in \mathcal {B}(\mathcal {H})^n\)  be a pure commuting \(n\) -tuple of operators which is regular at \(z=0\) . Then the following conditions are equivalent:

For some positive integer \(m \ge 1\) , \(X\)  satisfies the identity \((X^*X)^{-1} = (\oplus \Delta _X)|_{X^* \mathcal {H}}\) , where \(\Delta _X = \oplus \sum _{j=0}^{m-1} (-1)^j {m \atopwithdelims ()j+1} \sigma _X^j(I_H)\) ,

\(X\)  is unitarily equivalent to \(M_{\mathbf {z}} \in \mathcal {B}(H_m(\mathbb {B}_n, \mathcal {W}(X)))^n\) .


In particular, condition \((i)\)  implies that \(X\)  satisfies the wandering subspace property on \(\mathcal {H}\)  and there exists a unitary \(U:\mathcal {H}\rightarrow H_m(\mathbb {B}_n, \mathcal {W}(X))\)  defined by
\(Uh(\mathbf {z}) = \sum _{\alpha \in \mathbb {N}^n} \gamma _{\alpha } P_{\mathcal {W}(X)} L^{\alpha }h \hspace{2.84526pt} {\mathbf {z}}^{\alpha },\) 
where \(L:= (X^*X)^{-1} X^*\) . Moreover, the unitary \(U\)  follows the following intertwining property
\(UX_i = (M_{z_i} \otimes I_{\mathcal {E}}) U \quad (i \in \lbrace 1,\ldots ,n\rbrace ),\) 
and \(U(\mathcal {W}(X)) = \mathcal {W}(X)\)  (see [Theorem 3.5, [9]}] for details). We are now ready to state our main result based on these tuples.
Theorem 4.9 Let \(X \in \mathcal {B}(\mathcal {H})^n\)  be a pure commuting \(n\) -tuple of operators which is regular at \(z=0\)  and for some positive integer \(m \ge 1\)  satisfies \((X^*X)^{-1} = (\oplus \Delta _X)|_{X^* \mathcal {H}}\) . Let \(T\)  be a contraction on \(\mathcal {H}\)  such that \(T X_i = X_i T\) , for all \(i \in \lbrace 1,\ldots ,n\rbrace \) . Then the following are equivalent:
(i)
\(T\)  is a pure contraction on \(\mathcal {H}\) ,
(ii)
\(P_{\mathcal {W}(X)} T|_{\mathcal {W}(X)}\)  is a pure contraction on \(\mathcal {W}(X)\) .
Note that \((i)\)  implies \((ii)\)  follows in a manner that is similar to the above results.
\((ii) \Rightarrow (i):\)  From equation (REF ), it follows that
\(UTU^* (M_{z_i} \otimes I_{\mathcal {E}}) = (M_{z_i} \otimes I_{\mathcal {E}}) UTU^* \quad (i \in \lbrace 1,\ldots ,n\rbrace ).\) 
This implies that there exists a contractive multiplier \(\Theta \in \mbox{Mult}_1(H_m(\mathbb {B}_n, \mathcal {W}(X)))\)  such that \(T = U^* M_{\Theta } U\)  (see Proposition 4.2, [13]}). Now assumption \((b)\)  implies that \(\mathcal {W}(X) \subseteq \ker A_{T}\) , and therefore, for any \(\eta \in \mathcal {W}(X)\)  we have
\(\lim _{n \rightarrow \infty } \Vert M_{\Theta }^{*n} U \eta \Vert  = \lim _{n \rightarrow \infty } \Vert U T^{*n} \eta \Vert  = \lim _{n \rightarrow \infty } \Vert T^{*n} \eta \Vert  =0.\) 
Thus, \(\mathcal {W}(X) = U \mathcal {W}(X) \subseteq \ker A_{M_{\Theta }}\)  and therefore, Theorem REF  implies that \(M_{\Theta }\)  is a pure contraction on \(H_m(\mathbb {B}_n, \mathcal {W}(X))\) . And by unitary equivalence, we get that \(T\)  is also a pure contraction on \(\mathcal {H}\) . This completes the proof.

Applications
Recently, the author proved that pair of commuting pure contractions with finite dimensional defect spaces admit pure isometric dilation (see [1]}). The aim of this section is to apply Theorem REF  to extend the above result for certain important tuples of commuting pure contractions appearing in recent papers [15]} and [16]}.
Let
\(\mathcal {X}^n(\mathcal {H}) := \lbrace (X_1,\ldots ,X_n) \in \mathcal {B}(\mathcal {H}): \Vert X_ih\Vert  \le 1, X_iX_j = X_jX_i, 1 \le i,j \le n\rbrace \) 
A tuple \(X \in \mathcal {X}^n(\mathcal {H})\)  is said to be pure if \(X_i\)  is a pure contraction for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .
With respect to the Szegö kernel \(\mathbb {S}_n(\mathbf {z}, \mathbf {w})\)  of the Hardy space \(H^2(n)\) , let
\(\mathbb {S}_n(\mathcal {H}): = \lbrace X \in \mathcal {X}^n(\mathcal {H}): \mathbb {S}_n^{-1}(X,X^*) \ge 0\rbrace .\) 
For the sequel, we need a special type of operator-valued multipliers called the Schur-Agler class on \(n\)  (see [17]}).
Definition 5.1 For a Hilbert space \(\mathcal {E}\) , the Schur-Agler class on \(n\)  (denoted by \(\mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\) ) is defined to be the collection of all \(\mathcal {B}(\mathcal {E})\) -valued bounded analytic function on \(n\)  for which there exists Hilbert spaces \(\lbrace \mathcal {H}_i\rbrace _{i=1}^n\)  and a unitary
\(U= \begin{bmatrix}A & B \\C & D\end{bmatrix}: \mathcal {E}\oplus (\underset{i=1}{\overset{n}{\oplus }} \mathcal {H}_i) \rightarrow \mathcal {E}\oplus (\underset{i=1}{\overset{n}{\oplus }} \mathcal {H}_i),\) 
such that
\(\Phi (\mathbf {z}) := A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n),\) 
where \(E(\mathbf {z}) := \oplus _{i=1}^n z_i I_{\mathcal {H}_i}\) .

It follows from a well-known computation that any element in \(\mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  is contractive.
Let us begin by observing a result that follows from [Theorem 4.2 and Proposition 4.1, [15]}]. For the sake of completeness we have included the proof of the result.
Lemma 5.2 ([15]},[16]}) 
Let \(T\)  be a contraction on a Hilbert space \(\mathcal {H}\)  for which there exists a Hilbert space \(\mathcal {E}\) , \(\Psi \in \mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  and an isometry \(\Pi :\mathcal {H}\rightarrow H_{\mathcal {E}}^{2}(n)\)  (where, \(n \ge 1\) ), such that \(\Pi T^* = M_{\Psi }^* \Pi \)  and \(H_{\mathcal {E}}^{2}(n) = \bigvee _{\mathbf {k} \in \mathbb {N}^n} M_{\mathbf {z}}^{\mathbf {k}} \Pi \mathcal {H}\) . If \(T\)  is a pure contraction on \(\mathcal {H}\) , then \(\Psi (0)\)  is a c.n.u. contraction on \(\mathcal {K}\) .

By definition, \(\Psi \in \mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  implies that there exist Hilbert spaces \(\lbrace  \mathcal {H}_i \rbrace _{i=1}^n\)  and a unitary \(U\)  of the form in equation (REF ) such that
\(\Phi (\mathbf {z}) = A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n).\) 
By the Nagy-Foias decomposition of a contraction there exists an orthogonal decompostion \(\mathcal {E}= \mathcal {E}_{0} \oplus \mathcal {E}_{1}\)  such that
\(A = \begin{bmatrix}A|_{\mathcal {E}_0} & 0 \\0 & A|_{\mathcal {E}_1}\end{bmatrix}: \mathcal {E}_0 \oplus \mathcal {E}_1 \rightarrow \mathcal {E}_0 \oplus \mathcal {E}_1,\) 
where, \(A|_{\mathcal {E}_{0}}\)  is a unitary and \(A|_{\mathcal {E}_1}\)  is a c.n.u. contraction. With respect to this decomposition of \(A\) , the transfer function \(\Psi (\mathbf {z})\)  can be decomposed in the following manner (see [Proposition 3.1, [16]}])
\(\Psi (z) = \begin{bmatrix}\Psi _{0}(\mathbf {z}) & 0 \\0 & \Psi _{1}(\mathbf {z})\end{bmatrix} \quad (\mathbf {z} \in n),\) 
where, \(\Psi _0(\mathbf {z}) = A|_{\mathcal {E}_0}\)  and \(\Psi _1(\mathbf {z}) = A|_{\mathcal {E}_1} + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \in \mathcal {S}_n(\mathcal {E}_1,\mathcal {E}_1)\) . The proof will be complete if we can show that \(\Pi \mathcal {H}\subseteq H_{\mathcal {E}_{1}}^{2}(n)\) , since then \(H_{\mathcal {E}}^2({n}) = \bigvee _{\mathbf {k} \in \mathbb {N}^n} M_{\mathbf {z}}^{\mathbf {k}} \Pi \mathcal {H}\subseteq H_{\mathcal {E}_{1}}^{2}(n) \subseteq H_{\mathcal {E}}^2({n})\)  will imply that \(\mathcal {E}_0 = \lbrace 0\rbrace \) . Now \(g \in H_{\mathcal {E}_0}^2(n)\)  implies that there exist \(h_m \in H_{\mathcal {E}_0}^2(n)\)  for all \(m \in \mathbb {N}\)  such that \(g = M_{\Psi }^m h_m\)  (since \(M_{\Psi _0}\)  is a unitary on \(H_{\mathcal {E}_0}^2(n)\) ). Thus, for \(f \in \Pi \mathcal {H}\)  we have
\(| \langle f,g \rangle | = | \langle f, M_{\Psi }^{m} h_m \rangle | = | \langle M_{\Psi }^{*m} \Pi h, h_m \rangle | = | \langle \Pi X^{*m}h, h_m \rangle | \le \Vert X^{*n}h\Vert  \Vert g\Vert ,\) 
Taking limit on both sides as \(m \rightarrow \infty \)  gives \(\langle f,g \rangle = 0\)  which implies that \(\Pi \mathcal {H}\perp H_{\mathcal {E}_0}^2({n})\) . This completes the proof.

Tuples associated to von Neumann inequality in polydisc
Let us begin by giving a brief introduction on the tuples of operators appearing in [15]}. A particular class of these tuples were first studied by Grinshpan et al. ([23]}) in connection to multivariate von Neumann inequality. For each \(i, p, q \in \lbrace 1,\ldots ,n\rbrace \)  such that \(p \ne q\) , let
\(\hat{X}_i := \lbrace X \in \mathcal {X}^n(\mathcal {H}): (X_1,\ldots ,X_{i-1},X_{i+1},\ldots ,X_n) \in \mathcal {X}^{n-1}(\mathcal {H})\rbrace .\) 
and
\(\mathcal {X}_{p,q}^n(\mathcal {H}) := \lbrace X \in \mathcal {X}^n(\mathcal {H}): \hat{X}_p, \hat{X}_q \in \mathbb {S}_{n-1}(\mathcal {H}) \text{ and } \hat{X}_p \text{ is pure}\rbrace .\) 
\(X \in \mathcal {X}_{p,q}(\mathcal {H})\)  is said to be of finite rank if
\(\mathcal {D}_{\hat{X}_{i}} := \overline{\mbox{ran}} \hspace{2.84526pt}\mathbb {S}^{-1}(\hat{X}_i,{\hat{X}_{i}}^*)^{\frac{1}{2}},\) 
is a finite dimensional subspace of \(\mathcal {H}\) , for \(i=p,q\) .
The following result on isometric dilation for these tuples was obtained in [Theorem 5.3, [15]}].
Theorem 5.3 
Let \(X=(X_1,\ldots ,X_n) \in \mathcal {X}_{p,q}^n(\mathcal {H})\)  be finite rank, then there exists a Hilbert space \(\mathcal {E}\) , inner multiplies \(\Phi _p,\Phi _q \in H_{\mathcal {E}}^{\infty }({n-1})\) 
and an isometry \(\Pi : \mathcal {H}\rightarrow H_{\mathcal {E}}^2({n-1})\)  such that
\(\Pi X_i^* = \left\lbrace \begin{array}{ll}M_{z_i}^* \Pi \quad (i \in \lbrace 1,\ldots ,n-1\rbrace  \setminus \lbrace p,q\rbrace ),\\M_{\Phi _i}^* \Pi \quad (i=p,q),\end{array}\right.\) 
where \(\Phi _p,\Phi _q\)  are inner polynomials of degree atmost one such that
\(\Phi _p(\mathbf {z}) \Phi _q(\mathbf {z}) = \Phi _q(\mathbf {z}) \Phi _p(\mathbf {z}) = z_p I_{\mathcal {E}} \quad (\mathbf {z} \in {n-1}).\) 
In other words, \(X\)  dilates to a \(n\) -tuple of commuting isometries
\((M_{z_1},\ldots ,M_{z_{p-1}}, M_{\Phi _p},M_{z_{p+1},} \ldots , M_{z_{q-1}}, M_{\Phi _q}, M_{z_{q+1}}, \ldots ,M_{z_n})\) 
on \(H_{\mathcal {E}}^2({n-1})\) .

A remarkable contribution of the above result lies in the fact that \(M_{\Phi _p},M_{\Psi _q}\)  are explicitly determined to be of Berger, Coburn and Lebow type isometries [25]}. In particular, there exists a collection \((\mathcal {E},U,P)\)  where, \(\mathcal {E}\)  is a Hilbert space, \(U\)  is a unitary and \(P\)  is an orthogonal projection such that
\(\Phi _p(\mathbf {z}) = (P+z_pP^{\perp })U^*; \quad \Phi _q(\mathbf {z}) = U(P^{\perp } + z_p P).\) 
Using Theorem REF , we can establish the following result on pure isometric dilation.
Theorem 5.4 Let \(X=(X_1,\ldots ,X_n)\)  be a tuple of commuting pure contractions which belongs to the collection \(\mathcal {X}_{p,q}^n(\mathcal {H})\)  and has finite rank. Then there exists a Hilbert space \(\mathcal {E}\)  such that \(X\)  dilates to a \(n\) -tuple of commuting pure isometries
\((M_{z_1},\ldots ,M_{z_{p-1}}, M_{\Phi _p},M_{z_{p+1},} \ldots , M_{z_{q-1}}, M_{\Phi _q}, M_{z_{q+1}}, \ldots ,M_{z_n})\) 
on \(H_{\mathcal {E}}^2({n-1})\) .

Using Lemma REF  and Theorem REF  we obtain that \(\Phi _p(0), \Psi _{q}(0)\)  are c.n.u. contractions on the finite dimensional space \(\mathcal {E}\) , which further implies that \(\Phi (0),\Psi (0)\)  are pure contractions on \(\mathcal {E}\)  (see [26]}). Now, Theorem REF  implies that \(M_{\Phi _p}, M_{\Psi _q}\)  are pure isometries. This completes the proof.

Tuples associated to commutant lifting in unit polydisc
In the same manner as above, let us begin with a brief overview of tuples of contractions appearing in [16]}. These tuples were first studied by Ball et al. in their influential paper [28]} on the commutant lifting problem in the polydisc. Given \(A \in \mathcal {B}(\mathcal {H})\) , a conjugate map \(C_A:\mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathcal {H})\)  is the completely positive map defined by
\(C_A(Y) := AYA^* \quad (Y \in \mathcal {B}(\mathcal {H})).\) 
Definition 5.5 \(\mathcal {P}^n(\mathcal {H})\)  is defined to be the collection of \(n\) -tuples of operators \(X \in \mathcal {X}^n(\mathcal {H})\)  that satisfy the following conditions:

\(\hat{X}_n \in \mathbb {S}_{n-1}(\mathcal {H})\) ;

There exist positive operators \(G_1,\ldots ,G_{n-1} \in \mathcal {B}(\mathcal {H})\)  (depending on \(X\) ) for which

\(I - X_n X_n^* = G_1 + \ldots + G_{n-1}\) ;

\( S_X(G_i):= \prod _{j=1; j \ne i}^{n-1} (I_{\mathcal {B}(\mathcal {H})} - C_{X_j}) G_i \ge 0\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .

are satisfied.


The following result on isometric dilation of these tuples was obtained by Barik et al. in [Theorem 4.4, [16]}].
Theorem 5.6 
If \(X \in \mathcal {P}_n(\mathcal {H})\)  is of finite rank, then there exists an inner multiplier \(\Phi \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n},\mathcal {D}_{\hat{X}_n})\)  and an isometry \(\Pi : \mathcal {H}\rightarrow H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\)  such that
\(\Pi X_i^* = \left\lbrace \begin{array}{ll}M_{z_i}^* \Pi \quad (i=1,\ldots ,n-1),\\M_{\Phi }^* \Pi \quad (i=n).\end{array}\right.\) 
In other words, every finite rank \(X \in \mathcal {P}_n(\mathcal {H})\)  dilates to \((M_{z_1},\ldots ,M_{z_{n-1}}, M_{\Phi })\)  on \(H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\) .

The precise structure of the multiplier \(\Phi \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n}, \mathcal {D}_{\hat{X}_n})\)  is the following:
let \(F_i^2 := S_{X}(G_i)\)  for \(i=1,\ldots ,n-1\)  and \(\mathcal {F}_i = \overline{ran} F_i\) . The unitary in our discussion is \(U:\mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i) \rightarrow \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i)\)  such that
\(U(D_{\hat{X}_n}h, F_1 X_1^*h,\ldots ,F_{n-1}X_{n-1}^* h) = (D_{\hat{X}_n}X_n^*h, F_1 h, \ldots , F_n h).\) 
Let
\(U= \begin{bmatrix}A & B \\C & D\end{bmatrix}: \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i) \rightarrow \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i),\) 
and \(E(\mathbf {z}) := \oplus _{i=1}^{n-1} z_i I_{\mathcal {F}_i} \in \mathcal {B}(\oplus _{i=1}^{n-1} \mathcal {F}_{i})\) . Then \(\Phi (\mathbf {z})\)  has the realization in the following manner,
\(\Phi (\mathbf {z}) := \tau _{U}(\mathbf {z}) = A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n),\) 
Using Theorem REF , we obtain the following result on pure isometric dilation.
Theorem 5.7 Let \(X=(X_1,\ldots ,X_n)\)  be a tuple of commuting pure contractions which belongs to the collection \(\mathcal {P}_n(\mathcal {H})\)  and has finite rank. Then there exists a pure inner multiplier \(\Psi (z) \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n},\mathcal {D}_{\hat{X}_n})\)  such that \(X\)  dilates to a \(n\) -tuple of commuting pure isometries \((M_{z_1}, \ldots , M_{z_{n-1}}, M_{\Phi })\)  on \(H_{\mathcal {D}_{\hat{X}_{n}}}^2({n-1})\) .

Using Lemma REF  and Theorem REF  we obtain that \(\Phi (0)\)  is a c.n.u. contraction on the finite dimensional Hilbert space \(\mathcal {D}_{\hat{X}_{n}}\)  and therefore, \(\Phi (0)\)  is a pure contraction on \(\mathcal {D}_{\hat{X}_{n}}\) . Now, Theorem REF  shows that \(M_{\Phi }\)  is a pure isometry on \(H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\) . This finishes the proof.

Acknowledgements
The author would like to thank Prof. B.K. Das for useful discussions and he is grateful to Prof. Kunyu Guo for a correspondence regarding Proposition REF . He would also like to thank Prof. Gautam Bharali for going through the initial drafts of this article and providing important feedback. The author is supported by DST-INSPIRE Faculty Fellowship No. - DST/INSPIRE/04/2019/000769.
